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Kaiden Wilkins 2022-05-07 Answered
Prove that if $a,b,c\in \mathbb{R}$ are all distinct, then $a+b+c=0$ if and only if $\left(a,{a}^{3}\right),\left(b,{b}^{3}\right),\left(c,{c}^{3}\right)$ are collinear.
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Ariella Bruce
Step 1
The first part of your proof is enough because you can proceed by equivalence

Here is an alternate proof:
Using the classical alignment criteria for 3 points $\left({x}_{k},{y}_{k}\right)$ for k=1,2,3 which is
$|\begin{array}{ccc}{x}_{1}& {x}_{2}& {x}_{3}\\ {y}_{1}& {y}_{2}& {y}_{3}\\ 1& 1& 1\end{array}|=0$
This means that we have to show that

But this is very easy because the determinant can be factorized in the following way:
$\left(a-c\right)\left(b-a\right)\left(b-c\right)\left(a+b+c\right),$
knowing that a,b,c are all different.
In fact, I just realized that I had already answered a similar question here... with two proofs, this one and another one based on a third degree equation with no term in ${x}^{2}$